CHAPTER 10leeds-faculty.colorado.edu/.../FNCE4030-Fall-2012-ch10.pdf · 2012-10-29 · INVESTMENTS...

INVESTMENTS | BODIE, KANE, MARCUS

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

CHAPTER 10

Arbitrage Pricing Theory

and

Multifactor Models of Risk and Return

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10-2

Single Factor Model

• Returns on a security come from two

sources:

– Common macro-economic factor

– Firm specific events

• Possible common macro-economic

factors

– Gross Domestic Product Growth

– Interest Rates

– What else?


10-3

Single Factor Model Equation

ri = Return on security

βi= Factor sensitivity or factor loading or factor

beta

F = Surprise in macro-economic factor

(F could be positive or negative but has

expected value of zero)

ei = Firm specific events (zero expected value)

(uncorrelated with each other and with F)

( )i i i ir E r F e


10-4

Multifactor Models

• Use more than one factor in addition

to market return. Examples include:

– gross domestic product

– expected inflation

– interest rates

– …

• Estimate a beta or factor loading for

each factor using multiple regression.


10-5

Multifactor Model Equation

ri = Return for security i

βGDP = Factor sensitivity for GDP

βIR = Factor sensitivity for Interest Rate

ei = Firm specific events

iiIRiGDPii eIRGDPrEr


10-6

Multifactor SML Models

𝛽𝑖,𝐺𝐷𝑃 = Factor sensitivity for GDP

𝑅𝑃𝑖,𝐺𝐷𝑃 = Risk premium for GDP

𝛽𝑖,𝐼𝑅 = Factor sensitivity for Interest Rate

𝑅𝑃𝑖,𝐼𝑅 = Risk premium for Interest Rate

IRIRiGDPGDPifi RPRPrrE ,,


10-7

Interpretation

The expected

return on a

security is the

sum of:

1.The risk-free rate

2.The sensitivity to GDP

times the risk premium

for bearing GDP risk

3.The sensitivity to

interest rate risk times

the risk premium for

bearing interest rate

risk


10-8


1. Securities described with a Factor Model

2. There are enough securities to diversify away idiosyncratic risk

3. Arbitrage will disappear quickly

• Arbitrage when a zero investment portfolio has a sure profit

• No investment is required so investors can create large positions to obtain large profits


10-9


• Regardless of wealth

or risk aversion,

investors will want

an infinitely large

position in the risk-

free arbitrage

portfolio.

• In efficient

markets, profitable

arbitrage

opportunities will

quickly disappear.


10-10

APT & Well-Diversified Portfolios

rP = E (rP) + P F + eP

F = some factor

• For a well-diversified portfolio, eP

– approaches zero as the number of

securities in the portfolio increases

– and their associated weights decrease


10-11

Figure 10.1 Returns as a Function of the Systematic Factor

Well-diversified portfolio and single stock


10-12

Figure 10.2 Returns as a Function of the Systematic Factor: An Arbitrage Opportunity

Can the two portfolios co-exist?


10-13

Figure 10.3 An Arbitrage Opportunity


10-14

Figure 10.4 The Security Market Line


10-15

APT Model

• APT applies to well diversified portfolios

and not necessarily to individual stocks.

• With APT it is possible for some individual

stocks to be mispriced – not lie on the SML,

although APT must hold for most stocks (proof is difficult but reasoning can be illustrated)

• APT can be extended to multifactor

models.


10-16

APT and CAPM

APT

• Equilibrium means no arbitrage opportunities.

• APT equilibrium is quickly restored even if only a few investors recognize an arbitrage opportunity.

• The expected return–beta relationship can be derived without using the true market portfolio.

CAPM

• Model is based on an inherently unobservable “market” portfolio.

• Rests on mean-variance efficiency. The actions of many small investors restore CAPM equilibrium.

• CAPM describes equilibrium for all assets.


10-17

Multifactor APT

• Use of more than a single systematic factor

• Requires formation of factor portfolios

• What factors to choose?

– Factors that are important to

performance of the general economy

– What about firm characteristics?


10-18

Two-Factor Model

• The multifactor APT is similar to the

one-factor case.

• Each factor F has zero expected value

as it measures the surprise, not the

level.

• Also ei has zero expected value.

iiifii eFFrrEr 2211


10-19

Two (or multi)-Factor Model

• Track with diversified factor portfolios

• The factor portfolios track a particular

source of macroeconomic risk, but

are uncorrelated with other sources

of risk

• Each factor portfolio has 𝛽=1 for one of the

factors and 0 for all other factors


10-20

Where Should We Look for Factors?

• Need important systematic risk factors

– Chen, Roll, and Ross used industrial

production, expected inflation,

unanticipated inflation, excess return on

corporate bonds, and excess return on

government bonds.

– Fama and French used firm characteristics

that proxy for systematic risk factors.


10-21

Fama-French Three-Factor Model

• SMB = Small Minus Big (return of small in

excess of big firms, based on firm size)

• HML = High Minus Low (return of firms with high

book-to-market ratio, over those with low BtM)

• Are these firm characteristics correlated with

actual (but currently unknown) systematic risk

factors?

ittiHMLtiSMBMtiMiit eHMLSMBRr


10-22

The Multifactor CAPM and the APT

• A multi-index CAPM will inherit its risk

factors from sources of risk that a

broad group of investors deem

important enough to hedge

• The APT is largely silent on where to

look for priced sources of risk

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CHAPTER 10leeds-faculty.colorado.edu/.../FNCE4030-Fall-2012-ch10.pdf · 2012-10-29 · INVESTMENTS...

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